In the investigation of minimal machinery in reversible computing, we proved that each nondegenerate 2-state 3-symbol reversible logic element is logically universal. So far, a 2-state 4-symbol element called “rotary element” was shown to be logically universal in the framework of reversible logic element with memory. The main result in this paper not only improves the previous result with resp...

The higher order predictor (43) from [1] has to be modified in the degenerate case in order for the asymptotic convergence results from Lemma 4.6 and Theorem 4.7 to hold. More precisely equation (43) should be replaced by: where = 0 , if HLCP is nondegenerate 1 , if HLCP is degenerate. (2) The computational complexity results hold for the new algorithm, with slight modifications in the proof. C...

We consider a family of degenerate elliptic equations of the form div(∇F (∇u)) = f , where F ∈ C is a convex function which is elliptic outside a ball. We prove an excess-decay estimate at points where ∇u is close to a nondegenerate value for F . This result applies to degenerate equations arising in traffic congestion, where we obtain continuity of∇u outside the degeneracy, and to anisotropic ...

In this paper we study controllability properties of linear degenerate parabolic equations. Due to degeneracy, classical null controllability results do not hold in general. Thus we investigate results of 'regional null controllability', showing that we can drive the solution to rest at time T on a subset of the space domain, contained in the set where the equation is nondegenerate. keywords: l...

This paper presents an explicit formula that converts a triangular Bézier patch of degree n to a degenerate rectangular Bézier patch of degree n× n by reparametrization. Based on this formula, we develop a method for approximating a degenerate rectangular Bézier patch by three nondegenerate Bézier patches; more patches can be introduced by subdivision to meet a user-specified error tolerance. ...

Parametric oscillations are well studied phenomena, with applications in amplification, quantum optics, and quantum information processing. They can occur as a parameter of a system, such as the resonant frequency, is being modulated or “pumped” by an external field. A nonlinearity of the system can then transfer power from the pumping frequency to two frequencies called signal and idler. When ...

This article studies the asymptotic behavior of solutions to the damped, non-linear wave equation ü+ γu̇−m(‖∇u‖)∆u = f(x, t) , which is known as degenerate if the greatest lower bound for m is zero, and non-degenerate if the greatest lower bound is positive. For the nondegenerate case, it is already known that solutions decay exponentially, but for the degenerate case exponential decay has remai...

In Morse theory an isolated degenerate critical point can be resolved into a finite number of nondegenerate critical points by perturbing the totally degenerate part of the Morse function inside the domain of a generalized Morse chart. Up to homotopy we can admit pertubations within the whole characteristic manifold. Up to homotopy type a relative CW-complex is attached, which is the product of...

The nucleotide diversity (pi) in humans is studied by using published cDNA and genomic sequences that have been carefully checked for sequencing accuracy. This measure of genetic variability is defined as the number of nucleotide differences per site between two randomly chosen sequences from a population. A total of more than 75,000 base pairs from 49 loci are compared. The DNA regions studied...